On parsimonious edge-colouring of graphs with maximum degree three

TL;DR

This paper extends known results on the fraction of edges that can be optimally edge-colored in graphs with maximum degree three, identifying specific extremal graphs and using structural properties of minimal edge colorings.

Contribution

It generalizes a previous cubic graph result to all degree-3 graphs except one, and characterizes the extremal graphs achieving the bound.

Findings

The fraction of edges that can be edge-colored is at least 13/15 for most degree-3 graphs.

Exactly two graphs, including the Petersen graph, achieve this bound.

Structural properties of δ-minimum edge colorings are used to derive these results.

Abstract

In a graph $G$ of maximum degree $\Delta$ let $\gamma$ denote the largest fraction of edges that can be $\Delta$ edge-coloured. Albertson and Haas showed that $\gamma \geq 13/15$ when $G$ is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for which $\gamma = 13/15$. This extends a result given by Steffen. These results are obtained by using structural properties of the so called $\delta$-minimum edge colourings for graphs with maximum degree 3. Keywords : Cubic graph; Edge-colouring